Analysis of evidence-theoretic decision rules for pattern classification

The Dempster-Shafer (D-S) theory of evidence provides a convenient framework for modeling uncertainty in situations where the available evidence is limited or weak. Such situations typically arise in supervised pattern recognition when the training set is small or does not contain samples from all classes. In D-S theory, someone's feeling of uncertainty concerning a certain set of hypotheses is represented by a belief function, that generalizes the classical concept of probability distribution. Recently, several classiication methods have been proposed for determining a posterior belief function over a set of classes, given a pattern x to be classiied. However, the problem of making decisions based on a belief function and an arbitrary loss matrix is not simple, since conventional Bayes decision analysis assumes uncertainty to be quantiied by a probability distribution. This paper provides detailed analysis of several decision rules that can be based on a posterior belief function and a quantiication of the consequences of each action (assignment to a class or rejection). Diierent decision strategies are proposed, based on the concepts of upper, lower and pignistic expectation that generalize the usual concept of mathematical expectation underlying Bayes decision theory. The decision rules and the corresponding decision regions in feature space are analyzed under the assumptions of completeness and incompleteness of the training set. As expected, patterns situated close to class boundaries tend to be rejected according to the three decision rules studied. Additionally, the more conservative strategies of upper and pignistic expected loss minimization also lead to the rejection of \atypical" patterns situated far from training samples in feature space. When the training set is not complete, the available options are assignment to one of the known classes, assignment to the unknown class, and rejection. By imposing some restrictions on the loss matrix, it is shown that various conngurations of the decision regions can be induced by varying only two parameters corresponding to the costs of rejection and misclassiication in the unknown class, respectively. The analysis is illustrated using real data from a river quality monitoring application. Abstract The Dempster-Shafer theory provides a convenient framework for decision making based on very limited or weak information. Such situations typically arise in pattern recognition problems when patterns have to be classiied based on a small number of training vectors, or when the training set does not contain samples from all classes. This paper examines diierent strategies that can be applied in …